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Computation of the metaplectic kernel. (English) Zbl 0941.22019

Let \(K\) be a global field and \(G\) an absolutely simple simply connected algebraic group defined over \(K\). Let \(I=R/Z\). Clearly, \(I\) is a one-dimensional compact torus. For a given finite (possibly, empty) set \(S\) of places of \(K\), let \(A(S)\) denote the \(K\)-algebra of \(S\)-adeles. Let \(H^2(G(A(S)))\) be the second cohomology group of the group \(G(A(S))\) defined in terms of measurable cochains with values in \(I\) and let \(H^2(G(K))\) be the second cohomology group of \(G(K)\) defined in terms of abstract cochains with values in \(I\). Let \(M(S,G)\) be the kernel of the restriction map: \(H^2(G(A(S)))\to H^2(G (K))\). \(M(S,G)\) is called the metaplectic kernel. Let \(V\) be a finite set of places of \(K\). Let \(M_V(G)= \text{Ker} (H^2(G(V))\to H^2(G(K)))\). \(G/K\) is called to be special if it is the special unitary group of a nondegenerate hermitian form \(h\) over a noncommutative division algebra \(D\) with involution \(\tau\) of second kind (i.e., the center \(L\) of \(D\) is a quadratic extension of \(K=L^\tau)\). Conjecture \((U)\) is that if \(G/K\) is special, then for any finite set \(V\) of places of \(K\), \(M_V(G)= M_{V_0}(G)\), where \(V_0\) consists of all nonarchimedean places in \(V\). The authors give a precise computation of the metaplectic kernel as follows: Assume that \(G/K\) is special and that Conjecture \((U)\) holds for any finite set \(V\) of places of \(K\) not contained in \(S\). Then the metaplectic kernel \(M(S,G)\) is isomorphic to a subgroup of \(\widehat \mu(K)\), the dual of the group \(\mu(K)\) of roots of unity in \(K\).

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
20G30 Linear algebraic groups over global fields and their integers
12J20 General valuation theory for fields
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